Nonlinear optical rectification in the electric-field-biased parabolic quantum dots

~

Solid State Communications, Vol. 89, No. 12, pp. 1023-1027, 1993 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/94 $6.00 + .00

Pergamon

0038-1098(93)E0072-6 NONLINEAR OPTICAL RECTIFICATION IN THE ELECTRIC-FIELD-BIASED PARABOLIC QUANTUM DOTS Shi-Wei Gu* CCAST (World Laboratory), Beijing 100080, P.R. China and Kang-Xian Guo Department of Applied Physics and Institute of Condensed Matter Physics, Jiao Tong University, Shanghai 200030, P.R. China

(Received 15 July 1993 by B. Lundqvist) We report the calculations of the second-order optical susceptibilitynonlinear optical rectification Xo(2) of the electric-field-biased GaAs parabolic quantum dots. The large nonlinear optical rectification Xo(2) obtained in the electric-field-biased parabolic quantum dots is mainly due to the possibility of tuning independently the parabolic confinement frequency Wo, the biased electric field F and the angles (0 and ~b). The value of Xo(2) increases with increasing electric field F, decreases with increasing the parabolic confinement frequency Wo, and depends on the direction of the electric field (0 and ~b). Comparing with the results obtained in a Q2D system, we show that, as the electronic confinement is increased with the reduction of the dimensionality, the nonlinear optical rectification will be more pronounced.

1. INTRODUCTION THE DEVELOPMENT of experimental techniques such as chemical vapor deposition, liquid-phase epitaxy, and molecular-beam epitaxy have led to the fabrication of many quantum-weU structures with dimensions comparable to the electronic de Broglie wavelength. Due to their small size these structures present some physical properties such as optical and electronic transport characteristics that are quite different from those of the bulk semiconductor constituents [1, 2]. It is expected that these characteristic will be more pronounced as the electronic confinement is increased with the reduction of the dimensionality. Structures produced by ultrathin-film growth are inherently two dimensional, and thus experimental and theoretical investigations have largely been devoted to the structures in which only the carrier momentum normal to the interfaces is * Also at: Department of Applied Physics and Institute of Condensed Matter Physics, Jiao Tong University, Shanghai 200030, P.R. China.

quantized. Recent advances in microfabrication technology [3-5] have allowed the fabrication of structures with quantum confinement to one dimension ["quantum-well wires" (QWWs)] and have initiated intriguing investigations into one-dimensional physics. It is expected that the fabrication of semiconductor with quantum confinement to zero dimension ["quantum dots" (QD)] will show exotic electronic behavior. With recent advances in microfabrication, it is now possible to make zero-dimensional quantum dot nanostructures in which carriers are confined in all three dimensions. The electronic states of these structures resemble giant artificial atoms. Despite the fact that substantial experimental and theoretical progress has been achieved in understanding the relevant physical mechanisms in quantum dots, much of the experimental data is strongly influenced by the nonideal samples available for study: e.g. size distributions of the quantum dots, nonideal crystallization, possible impurities, traps, etc. It is expected that the optical nonlinearities are more sensitive to nonsquare quantum well shapes than to square well shapes [6]. In comparison to

1023

NONLINEAR OPTICAL RECTIFICATION IN QUANTUM DOTS

1024

square QWs, parabolic quantum wells have been shown to exhibit properties such as a nearly uniformly spaced density of states for the electrons and holes [7]. As even-order susceptibilities vanish in structures with inversion symmetry, finite secondorder susceptibilities can only be observed if the symmetry of the conduction-band potential is broken through either the growth of an asymmetric well or the application of an external bias field. The nonlinear optical rectification in the electric-fieldbiased parabolic quantum wells has been worked out recently [8]. In this paper, we report here on the calculations of the nonlinear optical rectification of the parabolic quantum dots biased with an external electric field F. The purposes of this paper are the following. In Section 2, we shall present a compact description of the density-matrix formalism which leads to a simple expression of nonlinear optical rectification in the electric-field-biased parabolic quantum dots. In Section 3, we will give our results and discussions. We find that the dipolar matrix elements (#021~Ax+ #021yAy+ #gl~Az) increase with increasing the electric field F, but decrease with increasing the parabolic confinement frequency w0 (see Figs. 2-4). We also show that, for a given confinement frequency w0 and electric field strength F, we obtain the maximum of 2 the dipolar matrix elements (#02txAx+#01rAy+ #021~Az) for 0 = ~ = 45 ° (see Figs. 2-4. From Fig. 5, we know that, for the same parameters, the nonlinear optical rectification X~2) in our model (0 = 0 = 45 °) is over 0.70 times larger than the results obtained in the Q2D system [8]. 2. THEORY Let us consider an electric-field-biased parabolic quantum dot in which electrons are confined in all three dimensions. The parabolic quantum dot exists in the GaAs region of the superlattice. The effectivemass Hamiltonian for the electron is given by

"=

h2 [0 2 0 2 0b_~z2] -2m---;L +O7 + q_l ~*

.2, 2__ 1 ~*

,_,

,2:

z

~

F

Quantum 0 dot ~

//

I f

~ y

-'

x

Fig. 1. Schematic diagram showing the direction of the biased electric field F in a parabolic quantum dot. origin is chosen at the center of the quantum dot. The eigenfunctions ~.,~(r) and the eigenenergies e.,k are solutions of the Schr6dinger equation H~.,k(r ) = e.,k~.,k, and are given by ffJ.,k(r) = ~.x(x)~.y(y)O).,(z),

(2)

and

e.,k=E.x+E.

+ E . ,.

(3)

Here, ~,~ and E~x are, respectively, the envelope wave function and the energy of the nxth subband, solutions of the one-dimensional Schr6dinger equation

HxCb.x(x) = [email protected](x),

(4)

where Hx is the x part of the Hamiltonian H in equation (I), i.e., h2

Hx

0 2

1 ~*

.2~2

2m, Ox 2 ~-U" ~o~ +qFxsinOcosfb.

• ~y and Ene are, respectively, the envelope wave function and the energy of the nyth subband, solutions of the one-dimensional Schr6dinger equation n y ~ n y ( y ) = Eny~ny(Y),

....

(5)

where Hy is the y part of the Hamiltonian H in equation (1), i.e.,

.2 2

~',, ~oy -r i,,, ~o~ + qF[xsinOcos~b

+ y sin 0 sin q~+ zcos 0],

Vol. 89, No. 12

(1)

where we have assumed that w0 = WOx = Woy = WOz, m* is the effective mass of the electron in GaAs (m* = 0.067m, m is the mass of bulk GaAs). ~1m*wox2 2, 1 ~ * .2 ,2 1 * 2 2 i,,, ~0y and irn w0z are, respectively, the parabolic confining potential in the x direction, y direction and z direction. F is the strength of the biased electric field along the direction (0 and ~b) shown in Fig. 1. The

Hy=

fi2

02

- - 1...*. 2 , , 2

2m. OyZ-V~m wo~, +qFysinOsinok

• .= and E., are, respectively the envelope wave function and the energy of the nzth subband, solutions of the one-dimensional Schr6dinger equation

Hz~.,(z) = E.~b.~(z),

(6)

where H~ is the z part of the Hamiltonian H in

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N O N L I N E A R OPTICAL R E C T I F I C A T I O N IN Q U A N T U M DOTS

equation (1), i.e., 1~2 0 2

Hz =

1025

dots 3

2) = , q

. I - - * .2_2

Ps: 2 A

2

2m, Oz2 r i m wo~ +qFzcos/9.

× The subband energy dispersion curves are given in equation (3). When the wavelength of the laser is far off-resonant from any electronic transitions, so that one can safely ignore the effects of excitons and multiple photon resonance, the physics then becomes quite simple. For semiconductor clusters (quantum dots) embedded in a dielectric medium, the major factor to be considered is the local field effect. For very small clusters, one has to consider the contribution of dipole-dipole interaction, that is, the reaction field, to the total internal field [9]. Let us consider the system described by the Hamiltonian H in equation (1). The system is excited by an internal electromagnetic field E ( t ) = Ecoswt. We suppose that the internal electromagnetic field be along the direction of the biased electric field F. The evolution of the density matrix equation with intraband relaxation becomes

OpO._ 1 [Hx _ qxE(t) sin 0cos ~b,p]/j Ot ih

1 [H y - qyE( t) sin Osin ~, p] O +~--~ 1 H - qzE(t) cos/9, ply. - Fo.(p - p(°))ij,

(7)

where p(0) is the unperturbed density matrix. For the sake of simplicity, we assume only two different values of the relaxation rates: Fl = 1/Tl for i = j is the diagonal relaxation rate and P2 = 1 / T2 for i # j is the off-diagonal relaxation rate. The electronic polarization of the electric-fieldbiased parabolic quantum dots due to the internal field E(t) = E c o s ~ t can be expressed as . . (2)(w)E2e-2iut + c.c. P(t) = eoX(t)(w)Ee -i~t + ~0;¢~ + e0X(2)(o.))E2,

(8)

where eo is the vacuum permittivity, X0)(w) and (2) X2~(w) are the linear and second-order optical susceptibility coefficients, X(02) is the optical rectification coefficient. Using the same density matrix approach as [8], we have calculated the expression (2) of Xo for the electric-field-biased parabolic quantum

g,[1 + r d r , ] -

+ +

+ r

)[rdr, - 1] +

(9) '

where = (¢,xlXl¢ox),

Ax = qFsin 0cos ~b re.w2

× #01y = (¢~lylYlt~Oy),

my -

x

Az =

gFsin 0 sin m%v2

,

qFcos/9 01z =

m,

02

;

h is Planck's constant, q is the electronic charge, wol = (El - Eo) / h is Bohr's frequency; Ps is the density of electrons in the electric-field-biased parabolic quantum dots. From equation (9) we show different ways to enhance the X(o2). The geometrical factor (#21xAx+ #21ymy + #21zmz) , the doping concentration Ps, and the time constant product Tl T2. In our model, we can maximize the geometrical factor (#21xA x + #~lyAy+ #201zAz) by tuning independently the parabolic confinement frequency ~oo, the biased electric field F, and the angles (/9 and ~b). T2 is certainly governed by intrinsic mechanism such as electron-electron interaction or optical-phonon emission for an excitation energy without possibilities to act upon it. On the other hand, T1 is a population relaxation time and can be enhanced by storing the excited electrons on a metastable level. 3. RESULTS A N D DISCUSSIONS In this section, we calculate the optical rectification X~2) obtained in equation (9) numerically for various angles (0 and ~b), parabolic confinement frequencies ~v0 and applied electric field F. The parameters are as follows: m * = 0.067m, Ti = lps, T2 = 0.2ps, w = 1012s -1, and p, = 5 x 1024m -3. In Fig. 2, we plot the geometrical factor (#21xAx+ #~lyAy + #021zAz)with (a) ~b = 0 ° (or 90°), (b) ~b = 30 ° (or 60 °) and (c) ~b=45 ° . We can see that the geometrical factor increases for a given confinement frequency (w0 = 9.0 x 1014s -1) and angle (0 = 45°), as the electric-field strength F increases. The result is similar to what we obtain in a Q2D system [8]. The results for 0 ° and 90 ° are the same - - similarly, the results for 30 ° and 60 ° . The results for 45 ° are different from the previous two categories. From Fig. 2, it shows that, for a given confinement

N O N L I N E A R OPTICAL R E C T I F I C A T I O N IN Q U A N T U M DOTS

1026 50,

too = 9.0 x 1014s"1 0 = 45 °

40

40

2

b

0 = ~p =45 °

a

2o

30)

g

20

h

+

+

4-

eq~ 10

~ lO

0

0.5

1.0 F (107 V/m)

50 /[

too = 9.0 x 1014s "1

40

~ = 45°

"~ 30

f

+

e

~ 20 -

d

+

10 .-t

0

0.5

1.0

1.0

1.5

2.0

F (107 V/m)

frequency w0, electric field strength F and angle 0, we obtain the maximum of the geometrical factor for ~b = 45 °. From Fig.3, we can obtain the same results just by the commutation of the angles 0 and 4). In Fig. 4, we plot the geometrical factor (#2jxAx+ #21yAy + p,21zmz) for three different parabolic confinement frequencies: (i) w0 = 9.0 x 1014s - 1 , ( j ) u;0 = 1015 s -1, (k) ~0 = 1.2 x 1015 s -l. We can see that, as

~.

0.5

2.0

1.5

Fig. 2. The dipolar matrix elements (#21xAxq/~l,Ay + ~lzAz) is plotted vs the biased electric fiel~l strength F (w0 : 9.0 × l014 s -l, 0 = 45 °. (a), (b), and (c) refer to values for ~b : 0 ° (or 90°), 30 ° (or 60°), and 45 °, respectively.

~

t

C

4-

~

Vol. 89, No. 12

50

1.5

2.0

F (107 V/m) 2 Fig. 3. The dipolar matrix elements (/z01xAx+ 2 2 #01yA>, +#0),Az) is plotted vs the biased electric field strength'F (w0 = 9.0 x 10 14 s- I , 4~ = 45 o). (d), (e) and ( f ) refer to values for 0 = 0 ° (or 90°), 30 ° (or 60°), and 45 °, respectively.

Fig. 4. The dipolar matrix elements (#~lxAx+ is plotted vs the biased electric field strength F (0 = ~b = 45°). It is plotted for three different confinement frequencies w0: (g) w0 = 9.0x 1014s-l; (h) OJo = 1015s-1 and (i) wO= 1.2 x 1015s -1.

#21ymy-~-~21zmz)

the confinement frequency w0 decreases, the geometrical factor increases for a given electric field F and angles (0 and 4~).This result is also similar to what we obtain in the Q2D system [8]. From Figs. 2-4, we show that the geometrical factor increases with increasing the applied electric field F, but decreases with increasing the parabolic confinement frequency ~o0. However, as we know, if the electric field F is too strong, it will break down the semiconductor clusters, that means there is a superior limit to F. Of course, there is also a limit to w0. From Figs. 2-4, we also see that the maximum of the geometrical factor (#021xAx + #21yAy + #~l~Az) occurs at 0 = ~b = 45 ° for a given electric field F and confinement frequency ~o0. In order to get large nonlinear optical rectification X~2.), we optimize F = 2 . 0 x 107Vm -1, w 0 = 3 . 6 x 1014s -l, and 0 = 0~ = 45 °. Figure 5 shows the nonlinear optical rectification X~2) as a function of photon energy obtained in our model. The huge enhancement in the theoretical values of the nonlinear optical rectification X~2) in our model comes from the possibility of independently tuning the electric field F, the confinement frequency ~0, and the angles (0 and 4~), In Fig. 5, the results are compared with those obtained in the Q2D system [8]. From comparing, we know that, using the same parameters, the nonlinear optical rectification X~2) in this Q0D system (when 0 = ~b = 45 °) is over 0.70 times larger than the results obtained in a Q2D system [8]. It means that the nonlinear optical (2) rectification X0 will be more pronounced as the

Vol. 89, No. 12

NONLINEAR OPTICAL RECTIFICATION IN QUANTUM DOTS

mainly from (ii) the possibility of tuning independently the confinement frequency ~0, the applied electric field F and the angles (0 and ~b). As expected, the nonlinear optical rectification will be more pronounced as the electronic confinement is increased with the reduction of the dimensionality. Finally, it is hoped that this paper would stimulate more experimental work which will be helpful in a understanding of the nonlinear optical rectification in the electric-field-biased parabolic quantum dots.

2.5

2.0

1.5

~=

I.C

0.50

1027

i

Acknowledgements - - This work was supported by the National Natural Science Foundation of China, Grant No. 69188006, and the National Educational Committee of China.

f

I

I

I

10

20

30

40

REFERENCES

k (l~m)

Fig. 5. The nonlinear optical rectification X~2) of the electric-field-biased GaAs parabolic quantum dots plotted as a function of the photon energy. T1 = 0.2 ps, T2 = l p s , w = 1012S-I, p s i 5 × 1024m-3, w0 = 3 . 6 x 1014s-l, F = 2 x 107Vm - , and 0 = 45 °. The results are compared with the results in a Q2D system (dashed line) in [8]. electronic confinement is increased with the reduction of the dimensionality. 4. CONCLUSIONS In this paper, we have calculated the nonlinear optical rectification X~2) of the electric-field-biased parabolic quantum dots. We show that the huge nonlinear optical rectification X~:) obtained in the structure are due to (i) the effect of the small effective mass (i.e., m* = 0.067m0 in GaAs quantum dots); but

1. 2. 3. 4. 5. 6. 7. 8. 9.

R. Tsu & L. Esaki, AppL Phys. Lett. 22, 562 (1973). T.C.L.G. Sollner, W.D. Goodhue, D.E. Tannewald & C.D. Parker, Appl. Phys. Lett. 43, 588 (1983). S. Luryi, Appl. Phys. Lett. 47, 490 (1985). T. Weft & B. Vinter, AppL Phys. Lett. 50, 1281 (1987). M. Jonson & A. Grimwajg, Appl. Phys. Lett. 51, 1729 (1987). A.C. Gossard, R.C. Miller & W. Wiegmann, Surf. Sci. 174, 131 (1986). R.C. Miller, A.C. Gossard, D.A. Kleinman & O. Munteanu, Phys. Rev. B29, 3740 (1984). K.X. Guo & S.W. Gu, Optical Nonlinearities in the Parabolic Quantum Wells with An Applied Electric Field (accepted by Phys. Rev. B, 1993). For a review, see C.J.F. Bottcher, Theory of Electric Polarization, Vols. 1 and 2, Elsevier, New York (1973).